You'd need a much weaker Senate, I think. Given how American institutions were designed rather than evolving, I don't see why they would deliberately make such a weak body, so perhaps instead America plumps for a unicameral Congress and the balance of power is different. Maybe merge the Senate and the Supreme Court into one entity, like how the House of Lords was (minus the religious aspect as the USA had no state religion) until recently.
While the US Constitution was designed in the sense that it its basic rules were all figured out in 1787, there's still a
lot of room for precedent and evolution -- particularly in the legislative process. Take for example the filibuster, of which there is no mention in the Constitution, but has become a dominant feature of how the Senate forms a check on the House.
In order to create a figure that operates like an American Prime Minister, you don't really need to do more than change the way the committee system operates. OTL the fact that the Speaker, as chair of the rules committee, uses his power to apportion committee chairmanships as a way to manage a governing coalition (a process not dissimilar from that used in the Senate, though with a good deal more "consensus" in the upper house) is largely a legacy of Henry Clay. Because of the shifting power of various Speakers, the ability of the Speaker to do so unilaterally is limited. The US House of Representatives under a strong speaker could easily claim far more power than it does OTL or did, under the Constitutional provision requiring all revenue bills originate with the lower house.
Neutering the Senate can also be accomplished through evolution and precedent, largely by altering the adoption of popular election for the Senate. Forestall the adoption of popular election and have a series of strong Speakers use the revenue provision to assert priority over the Senate; finally, replace popular election with some kind of override by the House over the Senate. In many ways, the composition, rules, and governing force of the Congress is the aspect of the US Constitutional Regime most suited to evolution over time.
It's important to remember as well that contemporary observes between 1875 - 1900 believed that the Speaker of the House would in future become the predominant office, in the wake of a series of powerful speakers and weak presidents. Teddy Roosevelt and Woodrow Willson are probably together responsible for the change. Modern Presidents use their veto far more than their 19th century predecessors did.
Yet another aspect of making the American system more parliamentary is to change the function and nature of the Cabinet. As OTL has evolved, the Cabinet is entirely a creature of the President's whim. Differing precedent (particularly between 1787 - 1800) could well have wrought a vastly different system. If Dem-Rep hadn't disliked Hamilton so much, Cabinet officers could well have non-voting seats in Congress and play a (larger) role in the Committee system. Had a version of the Tenure of Office Act (that in the 1860s sought to place limits on the President's ability to remove Congressional approved Cabinet officers) been adopted, Congress is again "in the driver's seat."
Finally, though, IMO the driver in the 19th Century away from Parliamentary / Congressional power and towards Presidential power was not so much a function of the US Constitution, but the scope of the Union. Congress met infrequently and frequent elections required a large number of Congressmen to travel incredibly long distances to either return home or to attend sessions. The relatively smaller size of the UK allowed Parliament to assemble relatively quickly. Even contrast the power of state legislatures and governors with Congress and the President: states with a smaller size, with legislatures able to meet more often, had less powerful governors. Whereas for almost six month out of any twelve in the 1800s, the President was nearly "the only game in town" inside of Washington DC.